Theorem tposfn2 5822

Description: The domain of a transposition. (Contributed by NM, 10-Sep-2015.)

Assertion

Ref	Expression
tposfn2	⊢ (Rel A → (𝐹 Fn A → tpos 𝐹 Fn ◡A))

Proof of Theorem tposfn2

Step	Hyp	Ref	Expression
1		tposfun 5816	. . . 4 ⊢ (Fun 𝐹 → Fun tpos 𝐹)
2	1	a1i 9	. . 3 ⊢ (Rel A → (Fun 𝐹 → Fun tpos 𝐹))
3		dmtpos 5812	. . . . . 6 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)
4	3	a1i 9	. . . . 5 ⊢ (dom 𝐹 = A → (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹))
5		releq 4365	. . . . 5 ⊢ (dom 𝐹 = A → (Rel dom 𝐹 ↔ Rel A))
6		cnveq 4452	. . . . . 6 ⊢ (dom 𝐹 = A → ◡dom 𝐹 = ◡A)
7	6	eqeq2d 2048	. . . . 5 ⊢ (dom 𝐹 = A → (dom tpos 𝐹 = ◡dom 𝐹 ↔ dom tpos 𝐹 = ◡A))
8	4, 5, 7	3imtr3d 191	. . . 4 ⊢ (dom 𝐹 = A → (Rel A → dom tpos 𝐹 = ◡A))
9	8	com12 27	. . 3 ⊢ (Rel A → (dom 𝐹 = A → dom tpos 𝐹 = ◡A))
10	2, 9	anim12d 318	. 2 ⊢ (Rel A → ((Fun 𝐹 ∧ dom 𝐹 = A) → (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ◡A)))
11		df-fn 4848	. 2 ⊢ (𝐹 Fn A ↔ (Fun 𝐹 ∧ dom 𝐹 = A))
12		df-fn 4848	. 2 ⊢ (tpos 𝐹 Fn ◡A ↔ (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ◡A))
13	10, 11, 12	3imtr4g 194	1 ⊢ (Rel A → (𝐹 Fn A → tpos 𝐹 Fn ◡A))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ^◡ccnv 4287  dom cdm 4288  Rel wrel 4293  Fun wfun 4839   Fn wfn 4840  tpos ctpos 5800

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-tpos 5801

This theorem is referenced by:  tposfo2  5823  tpos0  5830